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13    EXTENSIONS OF HERMITiAN OPERATORS       433

4 = 0 for all n. It follows that [/' is the Cayley transform of a closed Her-
mitian operator H' of defect (1,0), and consequently H" = -H'is a closed
Hermitian operator of defect (0, 1) (15.13.6). It can be shown that there exist
closed Hermitian operators of defect (m, n) for all choices of m and n in the
set N u {+00} (Problem 7).

In Chapter XXIII we shall see, by means of the theory of distributions,
how to describe the closure of the Hermitian operator H = z'D defined in
(15.13.2). We shall show that the defect of H** is (1, 1) if the interval I is
bounded, (1, 0) if I is bounded above but not below, (0, 1) if I is bounded
below but not above, and finally (0, 0) if /= R.

Theorem (15.13.6) generalizes as follows.

(15.13.10) For every complex number X such that S(X)>Q9 the subspace
E+(l) (resp. E~(A)) of dom(/f*) consisting of the solutions of H*  x = Ix
(resp. of H*  x = /be) is isomorphic to E# (resp. E#).

Since the subspace G~ + T(H) of T(H*) is closed in E x E (15.13.6), the
restriction H1 of H* to domCF^) = E# + dom(/f) is a closed operator. Put
A = \i + fv, where ^ and v are real and v > 0, and calculate (K^  A/) * jc||2
for some x = y + z e domC/^), where y e dom(# ) and z e E# . We have

and since H*  z =  iz, we have (H * y \ z) = (y \ H*  z) = i(y \ z). Now,
(15.13.10.1)      \\(H, - U)  *||2 = \\(H, - id) ' x\\2 + v2 W2

^ v2 ||*||2

by the preceding calculation and the hypothesis v > 0. If we put
F = Im^ - II), it follows that H^ - U is a bijection of domCHi) onto F,
and the inverse mapping (Hv  II)""1 is a continuous linear mapping of the
normed space F into E; since it is also a closed operator (because fli is closed),
we conclude (15.12.2) that the subspace F of E is closed in E. We shall show
that in fact F = E. We assert first that this is the case when A = i: for this we
have to prove that if ((H^ - //) * x \ z) = 0 for all x e dom^), then z must be en+1 for all n ^ 0,